Why Does Math Need Proofs?

Date: 03/24/2000 at 18:22:41
From: Lesa
Subject: Proofs
Why does math need to have proofs? I don't understand the importance
of them. Please explain.
Thank you for your time,
Lesa Schultz

Date: 03/25/2000 at 22:55:45
From: Doctor Peterson
Subject: Re: Proofs
Hi, Lesa.
We need proofs in math, first, because we want to be sure that what we
do is right. There are enough sources of error in our calculations,
from imprecise measurement to misunderstanding of the formulas we
should use, that it's important to make sure that our thinking doesn't
add more error. Proof just means checking our reasoning.
In math, unlike science or any other field, we CAN prove that what we
do is absolutely right. That's because math is not dependent on
partially known physical laws or unpredictable human behavior, but
simply on reason. In math, unlike the real world, we set the rules, so
we can know everything we need to know in order to be certain what
will happen. For example, we can define what we mean by addition, and
then prove that if we add b + a we will always get the same as a + b.
Since we can do it, we should take advantage of the possibility. Truth
is rare enough to value highly.
But the reason we really HAVE to prove things is that we can be easily
fooled. Some things that seem perfectly reasonable turn out to be
wrong. In fact, even if something is true whenever we try it, that
isn't enough to be sure that it always will be. The reason we can be
sure that a + b = b + a, for example, is not that we've always seen it
work that way, but that we can understand what is happening when we
add, and know that this rule is a natural result of the way addition
works. Often I see students trying to find a formula for some relation
(say, the number of diagonals in a polygon) by making a table and
looking for a pattern. Sometimes they find a formula that works for
the numbers they have; but if you add a line to the table, the formula
will no longer work. What they have to do is go back to the way the
table is made and see how a pattern will develop naturally. That's a
proof, and when you've done that, you KNOW it's right. You don't need
to guess and risk being fooled by a false pattern.
The Greeks were, as far as I know, the first to develop this love of
certainty. They saw that math was not just a tool they could use, but
a way to build a world of absolute truth, building one fact on another
so that they knew they were right. But they weren't perfect. They
originally built large parts of their geometrical thinking on the
assumption that any two lines could be compared by finding some unit
small enough that both lengths were whole-number multiples of that
unit; that is, all lines were assumed to be "commensurable." But it
was discovered that the diagonal of a square was incommensurable with
the side of the square - that is, the square root of two was
irrational. That shook them, and forced them to rethink their proofs,
since a lot of what they knew was based on a false assumption. They
were able to rebuild their geometry (Euclid's geometry incorporated
this rethinking) and as far as I know, nothing turned out to be wrong;
but the incident reinforced mathematicians' awareness of the
importance of really proving everything.
On the other hand, we can also be fooled in the other direction: there
are some things that are hard to believe without seeing a proof. For
example, I think it's hard to believe that the Pythagorean theorem
should always be true. I need a proof to convince me that I can always
use it and it will always work.
A different kind of proof can be useful in saving effort: the
existence proof. Sometimes it can take a lot of work to solve a
problem; a mathematician may first be able to prove whether a solution
exists, without having to do all the work of finding it. That can
either save us from bothering to try it, or allow us to work in
confidence, knowing there is an answer.
Not only mathematicians, but you yourself can benefit from learning to
do proofs. The skills you develop in learning to prove mathematical
statements are useful in many other areas of life. You learn logic,
which lets you recognize when a supposed "proof" (whether in math or
life) is flawed and shouldn't be believed. See our FAQ section on
False Proofs: http://mathforum.org/dr.math/faq/faq.false.proof.html .
You learn how to reason carefully and find links between facts. I
myself am a computer programmer, and though I don't prove theorems all
the time, I'm often checking whether a program will do what I expect,
and using those logical skills. Other people, from lawyers to
consumers, need to use logic in all sorts of ways.
You can find some other answers to your question by looking in our FAQ
under
Why study math?
http://mathforum.org/dr.math/faq/faq.why.math.html
and
Proofs
http://mathforum.org/dr.math/faq/faq.proof.html
The latter includes this link, which is worth looking at:
Proofs in Mathematics, Bogomolny
http://www.cut-the-knot.org/proofs/
If I haven't fully answered your question, feel free to write back.
- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/